X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FPartFunEquality.ma;h=c3b8f1c56b47a699698e8265edef6d718a6f1b90;hb=80110e17ef1d38d71473e9471ce15beddde663bb;hp=0ca5ba98a84281b6556bb80279dee6a375c6ce11;hpb=3199f2a8428b5d19a3a803c1b03d9f90e4120f74;p=helm.git diff --git a/helm/software/matita/contribs/CoRN-Decl/ftc/PartFunEquality.ma b/helm/software/matita/contribs/CoRN-Decl/ftc/PartFunEquality.ma index 0ca5ba98a..c3b8f1c56 100644 --- a/helm/software/matita/contribs/CoRN-Decl/ftc/PartFunEquality.ma +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/PartFunEquality.ma @@ -16,17 +16,15 @@ set "baseuri" "cic:/matita/CoRN-Decl/ftc/PartFunEquality". +include "CoRN.ma". + (* $Id: PartFunEquality.v,v 1.8 2004/04/23 10:00:59 lcf Exp $ *) (*#* printing Feq %\ensuremath{\approx}% #≈# *) -(* INCLUDE -Intervals -*) +include "reals/Intervals.ma". -(* INCLUDE -DiffTactics1 -*) +include "tactics/DiffTactics1.ma". (* UNEXPORTED Section Definitions. @@ -42,7 +40,7 @@ given predicate rather than to the predicate itself. The following definition makes this possible. *) -inline cic:/CoRN/ftc/PartFunEquality/subset.con. +inline "cic:/CoRN/ftc/PartFunEquality/subset.con". (*#* The core of our work will revolve around the following fundamental @@ -54,7 +52,7 @@ it is easier to consider a generic case%.}%. This file is concerned with proving the main properties of this equality relation. *) -inline cic:/CoRN/ftc/PartFunEquality/Feq.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq.con". (*#* Notice that, because the quantification over the proofs is universal, @@ -89,32 +87,32 @@ by [P] and [Q], respectively, the domains of [F] and [G]. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/I.var. +inline "cic:/CoRN/ftc/PartFunEquality/I.var". -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". -inline cic:/CoRN/ftc/PartFunEquality/G.var. +inline "cic:/CoRN/ftc/PartFunEquality/G.var". (* begin hide *) -inline cic:/CoRN/ftc/PartFunEquality/P.con. +inline "cic:/CoRN/ftc/PartFunEquality/P.con". -inline cic:/CoRN/ftc/PartFunEquality/Q.con. +inline "cic:/CoRN/ftc/PartFunEquality/Q.con". (* end hide *) -inline cic:/CoRN/ftc/PartFunEquality/R.var. +inline "cic:/CoRN/ftc/PartFunEquality/R.var". (*#* We start with two lemmas which make it much easier to prove and use this definition: *) -inline cic:/CoRN/ftc/PartFunEquality/eq_imp_Feq.con. +inline "cic:/CoRN/ftc/PartFunEquality/eq_imp_Feq.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_imp_eq.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_imp_eq.con". -inline cic:/CoRN/ftc/PartFunEquality/included_IR.con. +inline "cic:/CoRN/ftc/PartFunEquality/included_IR.con". (* UNEXPORTED End Equality_Results. @@ -132,7 +130,7 @@ Section Some_More. If two function coincide on a given subset then they coincide in any smaller subset. *) -inline cic:/CoRN/ftc/PartFunEquality/included_Feq.con. +inline "cic:/CoRN/ftc/PartFunEquality/included_Feq.con". (* UNEXPORTED End Some_More. @@ -158,17 +156,17 @@ the domain of [F]. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/I.var. +inline "cic:/CoRN/ftc/PartFunEquality/I.var". -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". (* begin hide *) -inline cic:/CoRN/ftc/PartFunEquality/P.con. +inline "cic:/CoRN/ftc/PartFunEquality/P.con". (* end hide *) -inline cic:/CoRN/ftc/PartFunEquality/bnd_away_zero.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_away_zero.con". (*#* If [F] is bounded away from zero in [I] then [F] is necessarily apart from zero in [I]; also this means that [I] is included in the domain of [{1/}F]. @@ -176,15 +174,15 @@ If [F] is bounded away from zero in [I] then [F] is necessarily apart from zero (* begin show *) -inline cic:/CoRN/ftc/PartFunEquality/Hf.var. +inline "cic:/CoRN/ftc/PartFunEquality/Hf.var". (* end show *) -inline cic:/CoRN/ftc/PartFunEquality/bnd_imp_ap_zero.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_imp_ap_zero.con". -inline cic:/CoRN/ftc/PartFunEquality/bnd_imp_inc_recip.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_imp_inc_recip.con". -inline cic:/CoRN/ftc/PartFunEquality/bnd_imp_inc_div.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_imp_inc_div.con". (* UNEXPORTED End Definitions. @@ -197,31 +195,31 @@ Boundedness away from zero is preserved through restriction of the set. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". -inline cic:/CoRN/ftc/PartFunEquality/P.var. +inline "cic:/CoRN/ftc/PartFunEquality/P.var". -inline cic:/CoRN/ftc/PartFunEquality/Q.var. +inline "cic:/CoRN/ftc/PartFunEquality/Q.var". -inline cic:/CoRN/ftc/PartFunEquality/included_imp_bnd.con. +inline "cic:/CoRN/ftc/PartFunEquality/included_imp_bnd.con". -inline cic:/CoRN/ftc/PartFunEquality/FRestr_bnd.con. +inline "cic:/CoRN/ftc/PartFunEquality/FRestr_bnd.con". (*#* A function is said to be bounded away from zero everywhere if it is bounded away from zero in every compact subinterval of its domain; a similar definition is made for arbitrary sets, which will be necessary for future work. *) -inline cic:/CoRN/ftc/PartFunEquality/bnd_away_zero_everywhere.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_away_zero_everywhere.con". -inline cic:/CoRN/ftc/PartFunEquality/bnd_away_zero_in_P.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_away_zero_in_P.con". (*#* An immediate consequence: *) -inline cic:/CoRN/ftc/PartFunEquality/bnd_in_P_imp_ap_zero.con. +inline "cic:/CoRN/ftc/PartFunEquality/bnd_in_P_imp_ap_zero.con". -inline cic:/CoRN/ftc/PartFunEquality/FRestr_bnd'.con. +inline "cic:/CoRN/ftc/PartFunEquality/FRestr_bnd'.con". (* UNEXPORTED End Away_from_Zero. @@ -263,23 +261,23 @@ partial functions. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/I.var. +inline "cic:/CoRN/ftc/PartFunEquality/I.var". (* UNEXPORTED Section Feq_Equivalence. *) -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". -inline cic:/CoRN/ftc/PartFunEquality/G.var. +inline "cic:/CoRN/ftc/PartFunEquality/G.var". -inline cic:/CoRN/ftc/PartFunEquality/H.var. +inline "cic:/CoRN/ftc/PartFunEquality/H.var". -inline cic:/CoRN/ftc/PartFunEquality/Feq_reflexive.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_reflexive.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_symmetric.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_symmetric.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_transitive.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_transitive.con". (* UNEXPORTED End Feq_Equivalence. @@ -293,31 +291,31 @@ Section Operations. Also it is preserved through application of functional constructors and restriction. *) -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". -inline cic:/CoRN/ftc/PartFunEquality/F'.var. +inline "cic:/CoRN/ftc/PartFunEquality/F'.var". -inline cic:/CoRN/ftc/PartFunEquality/G.var. +inline "cic:/CoRN/ftc/PartFunEquality/G.var". -inline cic:/CoRN/ftc/PartFunEquality/G'.var. +inline "cic:/CoRN/ftc/PartFunEquality/G'.var". -inline cic:/CoRN/ftc/PartFunEquality/Feq_plus.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_plus.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_inv.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_inv.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_minus.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_minus.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_mult.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_mult.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_nth.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_nth.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_recip.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_recip.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_recip'.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_recip'.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_div.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_div.con". -inline cic:/CoRN/ftc/PartFunEquality/Feq_div'.con. +inline "cic:/CoRN/ftc/PartFunEquality/Feq_div'.con". (*#* Notice that in the case of division we only need to require boundedness away from zero for one of the functions (as they are equal); thus the two last lemmas are stated in two different ways, as according to the context one or the other condition may be easier to prove. @@ -325,13 +323,13 @@ Notice that in the case of division we only need to require boundedness away fro The restriction of a function is well defined. *) -inline cic:/CoRN/ftc/PartFunEquality/FRestr_wd.con. +inline "cic:/CoRN/ftc/PartFunEquality/FRestr_wd.con". (*#* The image of a set is extensional. *) -inline cic:/CoRN/ftc/PartFunEquality/fun_image_wd.con. +inline "cic:/CoRN/ftc/PartFunEquality/fun_image_wd.con". (* UNEXPORTED End Operations. @@ -357,27 +355,27 @@ equality relation. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". (* begin hide *) -inline cic:/CoRN/ftc/PartFunEquality/P.con. +inline "cic:/CoRN/ftc/PartFunEquality/P.con". (* end hide *) -inline cic:/CoRN/ftc/PartFunEquality/Q.var. +inline "cic:/CoRN/ftc/PartFunEquality/Q.var". -inline cic:/CoRN/ftc/PartFunEquality/H.var. +inline "cic:/CoRN/ftc/PartFunEquality/H.var". -inline cic:/CoRN/ftc/PartFunEquality/Hf.var. +inline "cic:/CoRN/ftc/PartFunEquality/Hf.var". -inline cic:/CoRN/ftc/PartFunEquality/FNth_zero.con. +inline "cic:/CoRN/ftc/PartFunEquality/FNth_zero.con". -inline cic:/CoRN/ftc/PartFunEquality/n.var. +inline "cic:/CoRN/ftc/PartFunEquality/n.var". -inline cic:/CoRN/ftc/PartFunEquality/H'.var. +inline "cic:/CoRN/ftc/PartFunEquality/H'.var". -inline cic:/CoRN/ftc/PartFunEquality/FNth_mult.con. +inline "cic:/CoRN/ftc/PartFunEquality/FNth_mult.con". (* UNEXPORTED End Nth_Power. @@ -393,25 +391,25 @@ is included in the domain of [F]. %\end{convention}% *) -inline cic:/CoRN/ftc/PartFunEquality/a.var. +inline "cic:/CoRN/ftc/PartFunEquality/a.var". -inline cic:/CoRN/ftc/PartFunEquality/b.var. +inline "cic:/CoRN/ftc/PartFunEquality/b.var". -inline cic:/CoRN/ftc/PartFunEquality/Hab.var. +inline "cic:/CoRN/ftc/PartFunEquality/Hab.var". (* begin hide *) -inline cic:/CoRN/ftc/PartFunEquality/I.con. +inline "cic:/CoRN/ftc/PartFunEquality/I.con". (* end hide *) -inline cic:/CoRN/ftc/PartFunEquality/F.var. +inline "cic:/CoRN/ftc/PartFunEquality/F.var". -inline cic:/CoRN/ftc/PartFunEquality/incF.var. +inline "cic:/CoRN/ftc/PartFunEquality/incF.var". -inline cic:/CoRN/ftc/PartFunEquality/FNth_zero'.con. +inline "cic:/CoRN/ftc/PartFunEquality/FNth_zero'.con". -inline cic:/CoRN/ftc/PartFunEquality/FNth_mult'.con. +inline "cic:/CoRN/ftc/PartFunEquality/FNth_mult'.con". (* UNEXPORTED End Strong_Nth_Power.